Calculus, and what it really is.

I am not a mathematician, I am an engineer.
This isn't really a disclaimer, it's more of an explanation of why I wrote this article.
As an Engineer one is essentially forced to use Mathematics. But for engineers it's more of a tool than a persuit in and of itself.
I distinctly remember every mathematical epiphany I had, most of which related to exactly why I was doing something difficult that I didn't fully undertand.
Of course now I wish I had studied mathematics because it's beautiful, but as Napoleon said:
A man who is good at chess is a gentleman, but a master of chess is a wasted life
And when it comes to Mathemetics, this is every Engineer's mantra.

Anyway.. I wish to share with you an epiphany I was brought to be a particularly good teacher, many, many years ago.

Calculus and what it really is

There are three cars, a green one, a blue one and a red one. Here's a graph of the distance they cover per second.

The green car is not speeding up or slowing down, it's just doing 2 m/s (see the green line).
However the blue car is accelerating. Every second it gets 1 m/s faster. Its acelleration is 1 m/s/s
We can work it out without needing any fancy mathematics we can just see it. Every second it get's 1 m/s faster.
But if it were a bit harder to work out (maybe the line didn't pass through whole numbrers) then we can draw a triangle under the line and divide it's height (rise) by it's width (run). This gives us the slope of the line.
The slope of a line in a 'velocity' graph (𝑥 = metres / y = seconds) gives us 'acelleration'.
Interestingly the slope of the green line also gives us the acceleration of our first car, which is zero, there is no slope.

On the blue line we have another triangle further down which is only 1 high and 1 wide, but rise over run (slope) is still 1.
Does it matter how big the triangle is? No. Its height divided by it's width will always be 1.
10/10 = 1, pi/pi = 1, 0.1/0.1 = 1 etc.
The slope of the line of the function y = 𝑥 + 5 equals 1. And that is the 'derivative' of y = 𝑥 + 5.

As I said, it's easy to see this with a straight line but let's try it a different way.
We take a point on the blue line at 𝑥 = 5, we know that y = 𝑥 + 5, so when x is 4 y = 5 + 5 = 10
Now let's add a bit to x (in the case of the big triangle we added 5 to x).
To work out the new y we get (new y) = (x plus a bit (5)) + 5 Then we take another point when 𝑥 and y are a 'bit' bigger. so y + a bit = (x + 5) plus a bit.
We know that any change in y must equal (x + a change in x) + 5 But we've already established that it doesn't matter how big the triangle is.. it can be infinitely small.
So let's make

Start at the beginning

And here is all the background you need to understand what I just said.

Your average person doesn't really think about numbers but it turns out that there isn't just one type of number.
When you are beyond my level of mathematical ability you will learn 'group theory', and 'abstract' (or 'modern') algebra which is a branch of 'number theory' that requires you to understand what numbers actually are.
But here's a short introduction because you'll need it further down this post

Numbers

One of the first things we ever learn is how to count. And for most people the counting numbers (1, 2, 3, 4 etc.) are all we will ever use.
Counting up is easy because you never run out of numbers, but what happens if you count down, 4, 3, 2, 1 etc. then what?
Almost everyone knows that you get Zero next (although the concept of the number Zero is only around 3000 years old).
But what a lot of people don't know is that you can even carry on counting after zero into the 'Negative Numbers'
So counting down from 4 we get 4, 3, 2, 1, 0, -1, -2, -3, -4 etc.
This concept of having numbers that go right (upwards) and left (downwards) mathematicians usually represet with a Number Line

There are actually numbers that go up and down as well!? But I'll save them for a different article.

If you add or subtract any number on the line, you will arrive at annother number on the line.
So, adding means 'go right' and subtracting means 'go left':
1 + 3 = 4 (start on the number line at 1 and go right 3 spaces, gets you to 4)
2 - 3 = -1 (start on the number line at 2 and go left 3 spaces, and you get -1)
In actual fact "-" does not mean go left it really means 'change direction'. So if you're going right and you see a minus, then start going left.
If you then see another minus sign then change direction again. This sounds ridiculous but you'll see what I mean in a minute.

Even if we multiply we get another number on the line: 2 x 2 = 4 (better written as +2 x +2 = +4)
In terms of the number line mutiplication is just addition/subtraction only instead of using single spaces we use blocks of spaces.
2 x 2 = 4 means (starting at zero) go right two blocks of two spaces.

2 x -2 = -4 (Start at zero and go left by two blocks of two)
Ahh.. but What about -2 x -2, well when you have two minuses it means 'change direction twice'
Or in other words A minus times a minus is a positive. So -2 x -2 = 4
It's just a rule, remember it.

Division (Quotients or 'Rational Numbers' [ratio numbers])

You would think that division works the same way.. For example 6 ÷ 2 = 3 Means how many blocks of 2 are there in 6? For example.
But what if I say something like 1 ÷ 2 = ?. The result is not any number on the number line, it is something else, a fraction (posh name 'quotient') : ½
This type of number (quotients) are the second kind of number you'll find.

Irrational Numbers

It get's worse. A lot of people will have seen numbers like this : 2²
We can think of 'squaring' as just a shorthand notation for multiplication. 2² is just 2 x 2
But 2² actually indicates to us that a number should do something relating to itself (multiply by itself in this case).
And just like division is 'kind of' the opposite of multiplication (division), we have a 'kind of opposite' of 'squaring', which we call the 'square root'.
So let's take this √9 = 3. What, when multiplied by itself equals 9? Fair enough, we can express this as a fraction 9 ÷ 3 = 3 or 3 x 3 = 9
But what about this monster? : √2 = ?. What, when multiplied by itself equals 2, it isn't 1, because 1 x 1 = 1.

It turns out that we can't express this number as a fraction, or even a decimal. It's just not a ratio, it's 'Irrational'. It just is √2

Transcendence

But there are some numbers that you can't even write down using surds (√)
One famous example is this: How many 'diameters' of a circle are there in the 'circumference' of the circle.
I suppose we could write this as ^C⁄_D but that's not a number
There about 3.14 diameters of a circle it's circumference. Very nearly 22 ÷ 7, but you can't write it down exactly because it has an infinite amount of decimal places. It literally 'transcends' our ability to write it down.
So we give it a symbol, in this case: Π. But there are many other numbers that we need symbols to write down.

Just for fun, we learned above that a negative times a negative is a positive. So what is √-2. That is, what number multiplied by itself equals -2?
There is an answer but it's beyond the scope of this blog post.
And for the really inquisitive, what is ∛-27?
Thus ends the section about numbers.

Algebra

At some point using numbers becomes tedious.
Let's take this : A barrel has 10 apples in it, I put one of the apples in and 3 other people put (each the same amount) apples in. How many did each of the other people put in?

I'm going to use 'a' to stand for apples. We could write this down as 10a = 1a + (3 x ?a).
Ignoring that we're talking about 'apples' it might be: 10 = 1 + (3 x ?). but Mathematicians don't use the question mark they use letters (usually x).
They also don't use x to mean multiplication because you might confuse it with x as a letter. And even further, they don't write x they write 𝑥 So they would say 3𝑥 + 1 = 10 and then say something like : Solve for 𝑥
And just as a note: anything that has an = sign in it is an equation.
And one last note, the only important thing about equations is that they 'balance' so I can change an equation by doing something to one side, but I must also do something to the other side so that it all still balances.
For example 2 + 2 = 4 add 2 to one side, must add 2 to the other so 2 + 2 + 2 = 4 + 2, still balances.

Anyway 3𝑥 + 1 = 10 is pretty easy right? 𝑥 has to be 3!
What about this: Three people each put the same amount of apples in a barrel, then they go and have a beer. Two of them come back later and put the same amount of apples (as they did the first time) in again. At the end there are 10 apples in the barrel. How many apples did they put in each time?
Or: 3𝑥 + 2𝑥 = 10?
In other words on 5 occasions someone put an amount of apples in the barrel.
Or to put it another way we can 'collect the terms' : 3𝑥 + 2𝑥 = 5𝑥
So we know that 5𝑥 = 10. So 𝑥 must be 2!

Equations with more than one variable

Let's say that every time a lady puts some apples in the barrel she puts 3 in, and every time a man puts apples in a barrel he puts 2 in.
so : Three ladies and two men put apples in a barrel, so the barrel contains 3 x 3 + 2 x 2 = 13

Ok, 5 ladies and 3 men all place (a whole number of) apples in a barrel and at the end there are 37 apples. How many apples do ladies put in and how many do men put in?

This would be 5𝑥 + 3y = 37, how do we work this out?

Well let's re-arrange the equation by doing things to both sides:
5𝑥 + 3y = 37 is the same as (subtract 5𝑥 from both sides) 3y = 37 - 5𝑥
We could just start guessing values for 𝑥 and see if it works:
Let's try 𝑥 = 4. 3y = 37 - 5 x 4; gives 3y = 17, no good as 17 is not divisible by 3
Let's try 𝑥 = 1. 3y = 37 - 5 x 1; gives 3y = 32, no good as 32 is not divisible by 3
You can see that we might be here all year guessing values of 𝑥

Graphs

We know that when 𝑥 = 4 then y = ¹⁷⁄₃ (which is 5²⁄₃)
And when 𝑥 = 1 then y = ³²⁄₃ (which is 10²⁄₃) We could plot these as points on a graph and draw a line that goes through those points.
Then we could look to see if theres anywhere where that line passes through a point where both 𝑥 and y are whole numbers
Like this:

Who put what apples where now? — Fig.3 - Graph of 5𝑥 + 3y = 37

Here we can see that there are many (probably infinitely many) points where both the 𝑥 and y values are whole numbers.
However in almost all cases those whole numbers are negative numbers.
Unless people are taking apples out of the barrel, then negative numbers make no sense.
So we're looking for places where 𝑥 and y are positive numbers.

From this graph we can see two potential hits (highlighted with green circles) [𝑥=2, y=9] and [𝑥=5, y=4 ]
Try them :
5x2 + 3x9 = 37. Yes
5x5 + 3x4 = 37. Yes

Well.. I happen to know the ladies and gentlemen in question, and men always put 2 apples in while ladies always put 9 apples in.
But it's interesting that if it had been 5 and 4, there would still have been 37 apples in the barrel.

Triangles

Just for interest and for those of you that remember y = m𝑥 + c from school, here is a graph showing how we'd calculate slope and y intercept etc.
The important thing for us though, is that it shows a triangle which we can use to arrive at the same 'gradient' as we would using y = m𝑥 + c.
The 'slope' of a line is given by the triangle's rise over run.

Of course this graph doesn't really mean much. What is 𝑥? Well it's apples given by men.
It would be easier to understand if 𝑥 was 'metres' and y was seconds for example.
And now we're at the point where we can understand better what's going on in the section at the top of the post.